Robert Rosenthal died on
February 27, 2002, of a sudden heart attack. He was just 57, in the prime of
his professional life. He is missed and loved by the many friends, colleagues
and students who knew him. His incisive, constructive views helped develop many
new ideas, and even inspired directions of research that do not explicitly bear
his name. The fact that he never took received wisdom for granted was profoundly
nurturing, and he applied this ability not just to the research of others but
also to his own work. In examining his distinguished list of publications, one
is struck by a constant sense of unease with economic theory, and game theory
in particular, which led him to visit and revisit many of the outstanding problems
in the area - particularly those dealing with the question of rational behavior
- in different, eclectic ways.

Bob Rosenthal made fundamental
contributions to the way in which we think about game theory, some better known
than others. To take an example of a little-known contribution, Theorem 3 in
[8] is a formal statement of what is now widely known as the Revelation Principle.
Roger Myerson refers to this theorem in his own first paper on this subject
[12], indicating Bob's priority for this result. (Of course, the Revelation
Principle has been independently "discovered" more than once, and
Bob himself, seeing the result formally re-stated some years after his own rendition
of it, reportedly remarked, "Why on earth would anybody want to write that
down?")

What follows is a (necessarily)
selective account of some of Bob's contributions to economic theory. In [9]
Bob introduced a now-classical framework: random matching. To our understanding,
this is the first explicit and rigorous analysis of games played across "a
sequence of varying opponents." The paper reads as a reaction to the theory
of repeated games, observing correctly that many problems involve not repeated
play, but changing partners. One immediate and basic implication of this structure
is (or should be) that histories are imperfectly observed. Bob assumed that
only the most recent action taken by a freshly matched partner is observable.
With this assumption in hand, one may describe a (Markov) strategy by a rule
that maps one's immediate past actions, and that of the opponent, to an action
to be taken today. Together, these strategies induce a Markov process on societal
actions as a whole. Bob considered the steady states of such processes, against
which each individual Markov strategy forms a best response. [He recognized
that out-of-steady-state individual best responses are no longer Markov, thus
raising the very difficult question of convergence in this framework.]

Random matching is, of course,
widely used today. Indeed, Bob applied the idea himself in joint work with Henry
Landau [11]. This beautiful paper (one of the few for which Bob professed a
particular fondness) uses random matching to study the evolution of individual
reputations in bargaining, and the social influence of such reputations on individual
bargaining behavior. Call this influence a custom. If asymmetric outcomes (one
yields, the other is firm) are efficient in the sense of maximizing additive
surplus, as they often may be, then customs are potentially useful as a coordination
device. But there are many potential customs. Rosenthal and Landau study and
compare different customs that might persist in steady state, and compare the
efficiency gains they yield. Those interested in the evolution of social norms,
or more broadly in the social determinants of individual behavior, will benefit
from studying the methodology introduced in this paper.

In [10] Bob addressed the
paradoxical nature of backward induction in perfect information games. Of course,
he was not the only one to do so around this time, but he was possibly unique
(at the time) in refusing to budge from the complete information setup. In contrast
to incomplete information about types, Bob stressed the possibility that actions
may conform only in some probabilistic way to the best choice. There are many
provocative ideas raised in this paper. Among them is the introduction of the
beautiful and puzzling centipede game: not that it raised any new paradoxes
but that it displayed, in a minimal, elegant and forceful way, the nature of
the backward-induction problem. Little wonder that the centipede game is used
as a basic parable in textbooks, philosophical musings on the meaning of rationality,
as well as experiments.

A fresh reading of this
paper would be rewarding to many. In it, Bob advocates an approach which is
very close indeed to the popular quantal-choice model, in which the probability
of making a choice varies directly with the relative payoff advantage of that
choice, but may not be zero or 1. His beautiful insight is that such "trembles"
echo back, in an amplified way, as one recurses backwards through the game tree.
For instance, in the centipede game, a slight error at the very last stage actually
increases the relative payoff advantage of "passing" at the penultimate
stage, even though it may still be better to stop. This creates a slightly larger
tendency to pass at the penultimate stage. Now at the stage just anterior to
the penultimate stage it is even more profitable, relatively speaking, to pass.
As one inducts up the tree this must ultimately make "passing" the
best strategy, if the tree is long enough.

On another front, very early
in Bob's professional career, he expressed dissatisfaction with the way in which
cooperative game theory approaches the problems it seeks to tackle. (Of course,
Bob was not alone in making these points (for example, see [4] and W. Lucas,
"On solutions to n-person games in partition function form," Ph.D.
dissertation, University of Michigan, Ann Arbor, 1963). The attack [6,7] was
not along the popular lines that cooperative game theory relies too heavily
on binding agreements, which is no real criticism at all, but focused on the
very fundamentals of the theory: characteristic functions. Bob observed that
the construction of the characteristic function "implicitly assumes that
players expect other players actually to carry out threats which are injurious
to their own welfare." Of course, such a criticism has no value if there
are no externalities across agents, for then coalitions are effectively in isolation
anyway. But Bob's focus was on cases with externalities, and these early papers
realized, very clearly, that this shortcoming would be fundamental to cooperative
game theory. Only in recent literature has this issue been resurrected in a
fresh way.

In [2,3] Bob, with Douglas
Gale, investigated the consequences of types of bounded rationality exhibited
by "experimenters" and "imitators". In the context of an
example with one experimenter and a number of imitators, they explored in detail
the varied system dynamics that can be generated by the interaction of these
two behaviors. This example provides an interesting and elegant illustration
of how simple strategic "rules of thumb" can interact to produce complicated
dynamics, and also of the roles of substitutability and complementarity of actions
in games.

Motivated by the observation
that mixed strategies in games seem to have limited appeal in many practical
situations, in [5] Bob, with Roy Radner, explored the question of how far one
can get without using mixed-strategy equilibria. They proved that in games with
a finite number of players and moves, if each player observes a private signal
with an atomless distribution, which is independent of the observations and
payoffs of the other players, then for every mixed strategy equilibrium there
is a corresponding pure strategy equilibrium with the same expected payoffs
for each player. (In particular, this implies that every such game has a pure-strategy
equilibrium.) On the other hand, they gave examples of games that illustrate
the importance of the "independence assumption", as well as the importance
of the condition that each player have only finitely many moves. Hence the conditions
needed for the "purification" of mixed-strategy equilibria are surprisingly
strong. However, in [1] it is shown that conditions sufficient for the existence
of pure-strategy approximations to mixed-strategy equilibria are significantly
weaker.

We hope that this incomplete
account of Bob's research gives some idea of the range of his interests, critical
insights, and contributions. What this academic discussion cannot illustrate,
however, is Bob's many-layered web of rich and productive interactions with
researchers and students from all over the world. In the ultimate analysis,
and notwithstanding the deep nature of his own contributions, this ability to
interact was possibly Bob's defining feature as a scholar. His untimely death
will deprive economic theory of the important contributions that we would have
expected from him in the future, but more significantly, it has deprived us
- his friends, colleagues, and students - of Bob's insight, humor, and great
intelligence.